Notation for extracting value out of single element set
In some part of a document I am writing, I first define a set
$$
S : A to mathcal P(B) \
S(a) = { text{complicated expression here using $a$} }
$$
Later, I prove that $forall a,~exists x, ~S(a) = { x }$.
So, I want to now construct some notation for referring to the set with a single element ${ x } subset B$. Something like
$$unset(S(a))$$
where $unset$ is some valid definition / notation.
elementary-set-theory notation
asked 1 hour ago
Siddharth Bhat
2,8481918
add a comment |
In some part of a document I am writing, I first define a set
$$
S : A to mathcal P(B) \
S(a) = { text{complicated expression here using $a$} }
$$
Later, I prove that $forall a,~exists x, ~S(a) = { x }$.
So, I want to now construct some notation for referring to the set with a single element ${ x } subset B$. Something like
$$unset(S(a))$$
where $unset$ is some valid definition / notation.
elementary-set-theory notation
asked 1 hour ago
Siddharth Bhat
2,8481918
add a comment |
In some part of a document I am writing, I first define a set
$$
S : A to mathcal P(B) \
S(a) = { text{complicated expression here using $a$} }
$$
Later, I prove that $forall a,~exists x, ~S(a) = { x }$.
So, I want to now construct some notation for referring to the set with a single element ${ x } subset B$. Something like
$$unset(S(a))$$
where $unset$ is some valid definition / notation.
elementary-set-theory notation
asked 1 hour ago
Siddharth Bhat
2,8481918
In some part of a document I am writing, I first define a set
$$
S : A to mathcal P(B) \
S(a) = { text{complicated expression here using $a$} }
$$
Later, I prove that $forall a,~exists x, ~S(a) = { x }$.
So, I want to now construct some notation for referring to the set with a single element ${ x } subset B$. Something like
$$unset(S(a))$$
where $unset$ is some valid definition / notation.
elementary-set-theory notation
elementary-set-theory notation
asked 1 hour ago
Siddharth Bhat
2,8481918
asked 1 hour ago
Siddharth Bhat
2,8481918
asked 1 hour ago
Siddharth Bhat
2,8481918
asked 1 hour ago
Siddharth Bhat
2,8481918
asked 1 hour ago
Siddharth Bhat
2,8481918
2,8481918
add a comment |
add a comment |
3 Answers
3
active
oldest
votes
There is no commonly used notation specifically for this, so you shouldn't hesitate to just make up your own. One way you can express it with standard notation is with the symbol $bigcup$: if $S$ is a set, then $bigcup S$ denotes the union of all the elements of $S$, so $bigcup {x}=x$. This is kind of a "hack" though and is not something you can expect your readers to effortlessly understand. (Mathematically literate readers will figure it out, but it may take them a little work.)
Ultimately, the purpose of notation is to communicate, so you should pick your notation to communicate clearly and not be afraid to use words instead of notation if that would be clearer. I would probably recommend instead just writing something like:
We write $s(a)$ for the unique element of the set $S(a)$.
There's nothing special about the choice of $s$ as the function name for this; it's a reasonable choice to use to remind the reader that it is related to $S$ but there's nothing wrong with using a different name if you have a better reason.
answered 1 hour ago
Eric Wofsey
179k12204331
Thank you, I suppose I'll just do that then :) I was hoping for some existing notation, but oh well.
– Siddharth Bhat
1 hour ago
I have doubts regarding the validity of blindly using this notation. I think that implicitly the union must be over a collection of sets. If not then what the heck would $2cup {3}$ possibly mean?
– MPW
23 mins ago
@MPW Remember that $2={{},{{}}}$, so it's perfectly meaningful. :P (At least, if we're working in set theory.) More to the point, as long as the OP doesn't write "$bigcup x$" when $x$ isn't a set, they'll be fine. And actually, even outside of ZFC you can arguably make sense of all such expressions: e.g. "$2cup{3}$" is by definition ${x: xin 2$ or $xin{3}}$, and if $2$ has no elements this is just ${3}$. :P
– Noah Schweber
8 mins ago
add a comment |
It is just the union:
$$bigcup S(a) = bigcup{x} = x$$
Indeed, the union of a set $A$ is defined with $$c in bigcup A iff exists D in A text{ such that } c in D$$
For two sets we have the more familiar notation $A cup B = bigcup {A,B}$.
answered 1 hour ago
mechanodroid
26.1k62245
1
This is true but I wouldn't necessarily recommend actually using this notation--it is not particularly evocative of the desired meaning here and may confuse some readers.
– Eric Wofsey
1 hour ago
@EricWofsey Agreed, I wouldn't use any kind of "$text{unset}$" function. It would be best to state that $S(a) = {x}$ and then just use $x$.
– mechanodroid
1 hour ago
This is neat, I would not have thought of that! However, I'm accepting Eric's answer, since I believe it answers the question's spirit better, by suggesting some kind of wording.
– Siddharth Bhat
1 hour ago
add a comment |
Since it looks like you’re going to have to invent your own notation, why not just simply $hat{a}$ ? Why bring $S$ into the notation at all?
answered 31 mins ago
MPW
29.8k12056
add a comment |
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3 Answers
3
active
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3 Answers
3
active
oldest
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active
oldest
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active
oldest
votes
There is no commonly used notation specifically for this, so you shouldn't hesitate to just make up your own. One way you can express it with standard notation is with the symbol $bigcup$: if $S$ is a set, then $bigcup S$ denotes the union of all the elements of $S$, so $bigcup {x}=x$. This is kind of a "hack" though and is not something you can expect your readers to effortlessly understand. (Mathematically literate readers will figure it out, but it may take them a little work.)
Ultimately, the purpose of notation is to communicate, so you should pick your notation to communicate clearly and not be afraid to use words instead of notation if that would be clearer. I would probably recommend instead just writing something like:
We write $s(a)$ for the unique element of the set $S(a)$.
There's nothing special about the choice of $s$ as the function name for this; it's a reasonable choice to use to remind the reader that it is related to $S$ but there's nothing wrong with using a different name if you have a better reason.
answered 1 hour ago
Eric Wofsey
179k12204331
Thank you, I suppose I'll just do that then :) I was hoping for some existing notation, but oh well.
– Siddharth Bhat
1 hour ago
I have doubts regarding the validity of blindly using this notation. I think that implicitly the union must be over a collection of sets. If not then what the heck would $2cup {3}$ possibly mean?
– MPW
23 mins ago
@MPW Remember that $2={{},{{}}}$, so it's perfectly meaningful. :P (At least, if we're working in set theory.) More to the point, as long as the OP doesn't write "$bigcup x$" when $x$ isn't a set, they'll be fine. And actually, even outside of ZFC you can arguably make sense of all such expressions: e.g. "$2cup{3}$" is by definition ${x: xin 2$ or $xin{3}}$, and if $2$ has no elements this is just ${3}$. :P
– Noah Schweber
8 mins ago
add a comment |
There is no commonly used notation specifically for this, so you shouldn't hesitate to just make up your own. One way you can express it with standard notation is with the symbol $bigcup$: if $S$ is a set, then $bigcup S$ denotes the union of all the elements of $S$, so $bigcup {x}=x$. This is kind of a "hack" though and is not something you can expect your readers to effortlessly understand. (Mathematically literate readers will figure it out, but it may take them a little work.)
Ultimately, the purpose of notation is to communicate, so you should pick your notation to communicate clearly and not be afraid to use words instead of notation if that would be clearer. I would probably recommend instead just writing something like:
We write $s(a)$ for the unique element of the set $S(a)$.
There's nothing special about the choice of $s$ as the function name for this; it's a reasonable choice to use to remind the reader that it is related to $S$ but there's nothing wrong with using a different name if you have a better reason.
answered 1 hour ago
Eric Wofsey
179k12204331
Thank you, I suppose I'll just do that then :) I was hoping for some existing notation, but oh well.
– Siddharth Bhat
1 hour ago
I have doubts regarding the validity of blindly using this notation. I think that implicitly the union must be over a collection of sets. If not then what the heck would $2cup {3}$ possibly mean?
– MPW
23 mins ago
@MPW Remember that $2={{},{{}}}$, so it's perfectly meaningful. :P (At least, if we're working in set theory.) More to the point, as long as the OP doesn't write "$bigcup x$" when $x$ isn't a set, they'll be fine. And actually, even outside of ZFC you can arguably make sense of all such expressions: e.g. "$2cup{3}$" is by definition ${x: xin 2$ or $xin{3}}$, and if $2$ has no elements this is just ${3}$. :P
– Noah Schweber
8 mins ago
add a comment |
There is no commonly used notation specifically for this, so you shouldn't hesitate to just make up your own. One way you can express it with standard notation is with the symbol $bigcup$: if $S$ is a set, then $bigcup S$ denotes the union of all the elements of $S$, so $bigcup {x}=x$. This is kind of a "hack" though and is not something you can expect your readers to effortlessly understand. (Mathematically literate readers will figure it out, but it may take them a little work.)
Ultimately, the purpose of notation is to communicate, so you should pick your notation to communicate clearly and not be afraid to use words instead of notation if that would be clearer. I would probably recommend instead just writing something like:
We write $s(a)$ for the unique element of the set $S(a)$.
There's nothing special about the choice of $s$ as the function name for this; it's a reasonable choice to use to remind the reader that it is related to $S$ but there's nothing wrong with using a different name if you have a better reason.
answered 1 hour ago
Eric Wofsey
179k12204331
There is no commonly used notation specifically for this, so you shouldn't hesitate to just make up your own. One way you can express it with standard notation is with the symbol $bigcup$: if $S$ is a set, then $bigcup S$ denotes the union of all the elements of $S$, so $bigcup {x}=x$. This is kind of a "hack" though and is not something you can expect your readers to effortlessly understand. (Mathematically literate readers will figure it out, but it may take them a little work.)
Ultimately, the purpose of notation is to communicate, so you should pick your notation to communicate clearly and not be afraid to use words instead of notation if that would be clearer. I would probably recommend instead just writing something like:
We write $s(a)$ for the unique element of the set $S(a)$.
There's nothing special about the choice of $s$ as the function name for this; it's a reasonable choice to use to remind the reader that it is related to $S$ but there's nothing wrong with using a different name if you have a better reason.
answered 1 hour ago
Eric Wofsey
179k12204331
edited 59 mins ago
answered 1 hour ago
Eric Wofsey
179k12204331
answered 1 hour ago
Eric Wofsey
179k12204331
answered 1 hour ago
Eric Wofsey
179k12204331
179k12204331
Thank you, I suppose I'll just do that then :) I was hoping for some existing notation, but oh well.
– Siddharth Bhat
1 hour ago
I have doubts regarding the validity of blindly using this notation. I think that implicitly the union must be over a collection of sets. If not then what the heck would $2cup {3}$ possibly mean?
– MPW
23 mins ago
@MPW Remember that $2={{},{{}}}$, so it's perfectly meaningful. :P (At least, if we're working in set theory.) More to the point, as long as the OP doesn't write "$bigcup x$" when $x$ isn't a set, they'll be fine. And actually, even outside of ZFC you can arguably make sense of all such expressions: e.g. "$2cup{3}$" is by definition ${x: xin 2$ or $xin{3}}$, and if $2$ has no elements this is just ${3}$. :P
– Noah Schweber
8 mins ago
add a comment |
Thank you, I suppose I'll just do that then :) I was hoping for some existing notation, but oh well.
– Siddharth Bhat
1 hour ago
I have doubts regarding the validity of blindly using this notation. I think that implicitly the union must be over a collection of sets. If not then what the heck would $2cup {3}$ possibly mean?
– MPW
23 mins ago
@MPW Remember that $2={{},{{}}}$, so it's perfectly meaningful. :P (At least, if we're working in set theory.) More to the point, as long as the OP doesn't write "$bigcup x$" when $x$ isn't a set, they'll be fine. And actually, even outside of ZFC you can arguably make sense of all such expressions: e.g. "$2cup{3}$" is by definition ${x: xin 2$ or $xin{3}}$, and if $2$ has no elements this is just ${3}$. :P
– Noah Schweber
8 mins ago
Thank you, I suppose I'll just do that then :) I was hoping for some existing notation, but oh well.
– Siddharth Bhat
1 hour ago
Thank you, I suppose I'll just do that then :) I was hoping for some existing notation, but oh well.
– Siddharth Bhat
1 hour ago
I have doubts regarding the validity of blindly using this notation. I think that implicitly the union must be over a collection of sets. If not then what the heck would $2cup {3}$ possibly mean?
– MPW
23 mins ago
I have doubts regarding the validity of blindly using this notation. I think that implicitly the union must be over a collection of sets. If not then what the heck would $2cup {3}$ possibly mean?
– MPW
23 mins ago
@MPW Remember that $2={{},{{}}}$, so it's perfectly meaningful. :P (At least, if we're working in set theory.) More to the point, as long as the OP doesn't write "$bigcup x$" when $x$ isn't a set, they'll be fine. And actually, even outside of ZFC you can arguably make sense of all such expressions: e.g. "$2cup{3}$" is by definition ${x: xin 2$ or $xin{3}}$, and if $2$ has no elements this is just ${3}$. :P
– Noah Schweber
8 mins ago
@MPW Remember that $2={{},{{}}}$, so it's perfectly meaningful. :P (At least, if we're working in set theory.) More to the point, as long as the OP doesn't write "$bigcup x$" when $x$ isn't a set, they'll be fine. And actually, even outside of ZFC you can arguably make sense of all such expressions: e.g. "$2cup{3}$" is by definition ${x: xin 2$ or $xin{3}}$, and if $2$ has no elements this is just ${3}$. :P
– Noah Schweber
8 mins ago
add a comment |
It is just the union:
$$bigcup S(a) = bigcup{x} = x$$
Indeed, the union of a set $A$ is defined with $$c in bigcup A iff exists D in A text{ such that } c in D$$
For two sets we have the more familiar notation $A cup B = bigcup {A,B}$.
answered 1 hour ago
mechanodroid
26.1k62245
1
This is true but I wouldn't necessarily recommend actually using this notation--it is not particularly evocative of the desired meaning here and may confuse some readers.
– Eric Wofsey
1 hour ago
@EricWofsey Agreed, I wouldn't use any kind of "$text{unset}$" function. It would be best to state that $S(a) = {x}$ and then just use $x$.
– mechanodroid
1 hour ago
This is neat, I would not have thought of that! However, I'm accepting Eric's answer, since I believe it answers the question's spirit better, by suggesting some kind of wording.
– Siddharth Bhat
1 hour ago
add a comment |
It is just the union:
$$bigcup S(a) = bigcup{x} = x$$
Indeed, the union of a set $A$ is defined with $$c in bigcup A iff exists D in A text{ such that } c in D$$
For two sets we have the more familiar notation $A cup B = bigcup {A,B}$.
answered 1 hour ago
mechanodroid
26.1k62245
1
This is true but I wouldn't necessarily recommend actually using this notation--it is not particularly evocative of the desired meaning here and may confuse some readers.
– Eric Wofsey
1 hour ago
@EricWofsey Agreed, I wouldn't use any kind of "$text{unset}$" function. It would be best to state that $S(a) = {x}$ and then just use $x$.
– mechanodroid
1 hour ago
This is neat, I would not have thought of that! However, I'm accepting Eric's answer, since I believe it answers the question's spirit better, by suggesting some kind of wording.
– Siddharth Bhat
1 hour ago
add a comment |
It is just the union:
$$bigcup S(a) = bigcup{x} = x$$
Indeed, the union of a set $A$ is defined with $$c in bigcup A iff exists D in A text{ such that } c in D$$
For two sets we have the more familiar notation $A cup B = bigcup {A,B}$.
answered 1 hour ago
mechanodroid
26.1k62245
It is just the union:
$$bigcup S(a) = bigcup{x} = x$$
Indeed, the union of a set $A$ is defined with $$c in bigcup A iff exists D in A text{ such that } c in D$$
For two sets we have the more familiar notation $A cup B = bigcup {A,B}$.
answered 1 hour ago
mechanodroid
26.1k62245
answered 1 hour ago
mechanodroid
26.1k62245
answered 1 hour ago
mechanodroid
26.1k62245
answered 1 hour ago
mechanodroid
26.1k62245
26.1k62245
1
This is true but I wouldn't necessarily recommend actually using this notation--it is not particularly evocative of the desired meaning here and may confuse some readers.
– Eric Wofsey
1 hour ago
@EricWofsey Agreed, I wouldn't use any kind of "$text{unset}$" function. It would be best to state that $S(a) = {x}$ and then just use $x$.
– mechanodroid
1 hour ago
This is neat, I would not have thought of that! However, I'm accepting Eric's answer, since I believe it answers the question's spirit better, by suggesting some kind of wording.
– Siddharth Bhat
1 hour ago
add a comment |
1
This is true but I wouldn't necessarily recommend actually using this notation--it is not particularly evocative of the desired meaning here and may confuse some readers.
– Eric Wofsey
1 hour ago
@EricWofsey Agreed, I wouldn't use any kind of "$text{unset}$" function. It would be best to state that $S(a) = {x}$ and then just use $x$.
– mechanodroid
1 hour ago
This is neat, I would not have thought of that! However, I'm accepting Eric's answer, since I believe it answers the question's spirit better, by suggesting some kind of wording.
– Siddharth Bhat
1 hour ago
1
1
This is true but I wouldn't necessarily recommend actually using this notation--it is not particularly evocative of the desired meaning here and may confuse some readers.
– Eric Wofsey
1 hour ago
This is true but I wouldn't necessarily recommend actually using this notation--it is not particularly evocative of the desired meaning here and may confuse some readers.
– Eric Wofsey
1 hour ago
@EricWofsey Agreed, I wouldn't use any kind of "$text{unset}$" function. It would be best to state that $S(a) = {x}$ and then just use $x$.
– mechanodroid
1 hour ago
@EricWofsey Agreed, I wouldn't use any kind of "$text{unset}$" function. It would be best to state that $S(a) = {x}$ and then just use $x$.
– mechanodroid
1 hour ago
This is neat, I would not have thought of that! However, I'm accepting Eric's answer, since I believe it answers the question's spirit better, by suggesting some kind of wording.
– Siddharth Bhat
1 hour ago
This is neat, I would not have thought of that! However, I'm accepting Eric's answer, since I believe it answers the question's spirit better, by suggesting some kind of wording.
– Siddharth Bhat
1 hour ago
add a comment |
Since it looks like you’re going to have to invent your own notation, why not just simply $hat{a}$ ? Why bring $S$ into the notation at all?
answered 31 mins ago
MPW
29.8k12056
add a comment |
Since it looks like you’re going to have to invent your own notation, why not just simply $hat{a}$ ? Why bring $S$ into the notation at all?
answered 31 mins ago
MPW
29.8k12056
add a comment |
Since it looks like you’re going to have to invent your own notation, why not just simply $hat{a}$ ? Why bring $S$ into the notation at all?
answered 31 mins ago
MPW
29.8k12056
Since it looks like you’re going to have to invent your own notation, why not just simply $hat{a}$ ? Why bring $S$ into the notation at all?
answered 31 mins ago
MPW
29.8k12056
answered 31 mins ago
MPW
29.8k12056
answered 31 mins ago
MPW
29.8k12056
answered 31 mins ago
MPW
29.8k12056
29.8k12056
add a comment |
add a comment |
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